Skip to main content
SpringerLink
Log in

Generalized Weibull model-based statistical tensile strength of carbon fibres

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

A generalized Weibull model is presented for reliability characterization of single carbon fibres. This generalized distribution gives the probability of tensile failure of fibres as a function of length, surface area and volume of fibres, separately in different models using a statistically significant experimental data set. This model accounts for the unusual variation in the strength of the fibres as the physical parameters capture the severity of the most severe flaws in the ensemble of fibres. The experimental campaign involved analysing the diametric variation in each carbon fibre using photomicrographs. The micro-texture and morphology were further analysed using scanning electron microscopy images. An extensive characterization of the strength dependence on the fibre gage length was performed through “single-fibre uniaxial tensile tests” on several hundred carbon fibres of different grades. The goal of the study is to ensure the reliability of strength characterization of a single fibre, accounting for geometric irregularities and property variation. The present Weibull models are compared to assess their descriptive accuracy, that is, the “goodness-of-fit” in describing the observed statistical data as well their generalizability in predicting future values. The Kolmogorov–Smirnov (KS) test was used to ascertain the goodness of fit for the predicted Weibull plots. It was found that the descriptive adequacy of the models improved with the increase in the number of parameters used in the definition of the Weibull model. The conventional model failed the KS test for the most number of cases and hence was deemed unfit in terms of generalizability. Furthermore, the generalized Weibull model was found to have better descriptive accuracy and predictive power compared to the conventional model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Ladevèze, P., LeDantec, E.: Damage modelling of the elementary ply for laminated composites. Compos. Sci. Technol. 43(3), 257–267 (1992)

    Article  Google Scholar 

  2. Varna, J., Joffe, R., Talreja, R.: A synergistic damage-mechanics analysis of transverse cracking in \([\pm \theta /90_{4}]\) s laminates. Compos. Sci. Technol. 61(5), 657–665 (2001)

    Article  Google Scholar 

  3. Murari, V., Upadhyay, C.S.: Micromechanics based ply level material degradation model for unidirectional composites. Compos. Struct. 94(2), 671–680 (2012)

    Article  Google Scholar 

  4. Linhart, H., Zucchini, W.: Model Selection. Wiley & Sons, New York, USA (1986)

    MATH  Google Scholar 

  5. Pitt, M.A., Myung, I.J., Zhang, S.: Toward a method of selecting among computational models of cognition. Psychol. Rev. 109(3), 472 (2002)

    Article  Google Scholar 

  6. Weibull, W.: A statistical theory of the strength of materials. Ingeniörsvetenskapsakademiens Handlingar 151, 5–55 (1939)

    Google Scholar 

  7. Weibull, W.: A statistical distribution function of wide applicability. ASME J. Appl. Mech. Trans. ASME 18, 293–297 (1951)

    MATH  Google Scholar 

  8. Weibull, W.: A statistical distribution function of wide applicability (discussion). ASME J. Appl. Mech. Trans. ASME 18, 293–297 (1952)

    MATH  Google Scholar 

  9. Lavaste, V., Besson, J., Bunsell, A.R.: Statistical analysis of strength distribution of alumina based single fibres accounting for fibre diameter variations. J. Mater. Sci. 30, 2042–2048 (1995)

    Article  Google Scholar 

  10. Andersons, J., Joffe, R., Hojo, M., Ochiai, S.: Glass fibre strength distribution determined by common experimental methods. Compos. Sci. Technol. 62, 131–145 (2002)

    Article  Google Scholar 

  11. Zafeiropoulos, N.E., Baillie, C.A.: A study of the effect of surface treatments on the tensile strength of flax fibres: Part II. Application of Weibull statistics. Compos. Part A 38, 629–638 (2007)

    Article  Google Scholar 

  12. Xia, Z.P., Yu, J.Y., Cheng, L.D., Liu, L.F., Wang, W.M.: Study on the breaking strength of jute fibres using modified Weibull distribution. Compos. A 40, 54–59 (2009)

    Article  Google Scholar 

  13. Naito, K., Yang, J.M., Tanaka, Y., Kagawa, Y.: The effect of gauge length on tensile strength and Weibull modulus of polyacrylonitrile (PAN)- and pitch-based carbon fibers. J. Mater. Sci. 47, 632–642 (2012)

    Article  Google Scholar 

  14. Naito, K., Tanaka, Y., Yang, J.M., Kagawa, Y.: Tensile properties of ultrahigh strength PAN-based, ultrahigh modulus pitch-based and high ductility pitch-based carbon fibers. Carbon 46, 189–195 (2008)

    Article  Google Scholar 

  15. Dai, S.R., Piggott, M.R.: The strength of carbon and Kevlar fibres as a function of their lengths. Compos. Sci. Technol. 49, 81–87 (1993)

    Article  Google Scholar 

  16. Pan, N., Chen, H.C., Thompson, J., Inglesby, M.K., Khatua, S., Zhang, X.S., Zeronian, S.H.: The size effects on the mechanical behaviour of fibres. J. Mater. Sci. 32, 2677–2685 (1997)

    Article  Google Scholar 

  17. Wagner, H.D.: Stochastic concepts in the study of size effects in the mechanical strength of highly oriented polymeric materials. J. Polym. Sci. Part B: Polym. Phys. 27, 115–149 (1998)

    Article  Google Scholar 

  18. Wagner, H.D., Phoenix, S.L.: A Study of statistical variability in the strength of single aramid filaments. J. Compos. Mater. 18, 312–338 (1984)

    Article  Google Scholar 

  19. Wagner, H.D.: Statistical concepts in the study of fracture properties of fibres and composite. In: Friedrich, K. (ed.) Application of Fracture Mechanicas to Composite Materials. Elsevier Science Publisher B. V., Amsterdam (1989)

    Google Scholar 

  20. Karbhari, V.M., Wilkins, D.J.: Tensile failure load and fiber diameter as stochastic variables in the determination of fiber strength statistics. Reliab. Eng. Syst. Safe. 35, 73–76 (1992)

    Article  Google Scholar 

  21. Bartenev, G.M., Sidorov, A.B.: Statistical theory of the strength of glass fibers. Mekhanika Polimerov 2(1), 74–81 (1966)

    Google Scholar 

  22. Wilson, D.M.: Statistical tensile strength of Nextel \(^{{{\rm TM}}}\) 610 and Nextel\(^{{{\rm TM}}}\) 720 fibres. J. Mater. Sci. 32(10), 2535–2542 (1997)

    Article  Google Scholar 

  23. Curtin, W.A.: Tensile strength of fiber-reinforced composites: III. Beyond the traditional Weibull model for fiber strengths. J. Compos. Mater. 34, 1301–1332 (2000)

    Article  Google Scholar 

  24. Zhang, Y., Wang, X., Pan, N., Postle, R.: Weibull analysis of the tensile behavior of fibers with geometrical irregularities. J. Mater. Sci. 37, 1401–1406 (2002)

    Article  Google Scholar 

  25. Thomason, J.L.: On the application of Weibull analysis to experimentally determined single fibre strength distributions. Compos. Sci. Technol. 77, 74–80 (2013)

    Article  Google Scholar 

  26. van der Zwaag, S.: The concept of filament strength and the Weibull modulus. J. Test. Eval. 17(5), 292–298 (1989)

    Article  Google Scholar 

  27. Stoner, E.G., Edie, D.D., Durham, S.D.: An end-effect model for the single-filament tensile test. J. Mater. Sci. 29, 6561–6574 (1994)

    Article  Google Scholar 

  28. Rajendran, H., Mohite, P.M., Upadhyay, C.S.: Statistical Strength Determination of Carbon Fibres Using a Generalized Weibull Model. In: Proc. of 56th AIAA/ASME/SAE Struct., Struct. Dyn., Mater. Conf., Florida, USA (2015)

  29. Kirtay, S., Dispinar, D.: Effect of ranking selection on the Weibull modulus estimation. GU. J. Sci. 25(1), 175–187 (2012)

    Google Scholar 

  30. Chawla, K.K.: Fibrous Materials, pp. 15–51. Cambridge University Press, Cambridge, UK (1998)

    Book  Google Scholar 

  31. Morimoto, T.: A representative diameter for the Weibull scaling of variable diameter monofilament strength. Compos. A 34, 597–601 (2003)

    Article  Google Scholar 

  32. Watson, A.S., Smith, R.L.: An examination of statistical theories for fibrous materials in the light of experimental data. J. Mater. Sci. 20, 3260–3270 (1985)

    Article  Google Scholar 

  33. Gutans, J.A., Tamuzh, V.P.: Scale effect of the Weibull distribution of fibre strength. Mech. Comp. Mater. 6, 1107–1109 (1984)

    Google Scholar 

  34. Bergman, B.: On the estimation of the Weibull modulus. J. Mater. Sci. Lett. 3, 689–692 (1984)

    Article  Google Scholar 

  35. Bergman, B.: Estimation of Weibull parameters using a weight function. J. Mater. Sci. Lett. 5, 611–614 (1986)

    Article  Google Scholar 

  36. Myung, I.J.: Tutorial on maximum likelihood estimation. J. Math. Psychol. 47(1), 90–100 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wong, R.K.: Weibull distribution, iterative likelihood techniques and hydrometeorological data. J. Appl. Meteor. 16, 1360–1364 (1977)

    Article  Google Scholar 

  38. Huang, Y., Young, R.J.: Effect of fibre microstructure upon the modulus of PAN-and pitch-based carbon fibres. Carbon 33(2), 97–107 (1995)

    Article  MathSciNet  Google Scholar 

  39. Mochida, I., Yoon, S.H., Takano, N., Fortin, F., Korai, Y., Yokogawa, K.: Microstructure of mesophase pitch-based carbon fiber and its control. Carbon 34(8), 941–956 (1996)

    Article  Google Scholar 

  40. Kim, B.J., Eom, Y., Kato, 0., Miyawaki, J., Kim, B.C., Mochida, I., Yoon, S.H.: Preparation of carbon fibers with excellent mechanical properties from isotropic pitches. Carbon 77, 747–755 (2014)

    Article  Google Scholar 

  41. Paris, O., Loidl, D., Peterlik, H.: Texture of PAN-and pitch-based carbon fibers. Carbon 40(4), 551–555 (2002)

    Article  Google Scholar 

  42. ASTM C1557-03, Standard Test Method for Tensile Strength and Youngs Modulus of Fibers, Annual Book of ASTM Standards, Vol. 15.01, ASTM International: West Conshohocken, PA, 787796 (2010)

  43. ASTM D3379-89e1, Standard Test Method for Tensile Strength and Youngs Modulus for High-Modulus Single-Filament Materials, Annual Book of ASTM Standards, Vol. 15.01, ASTM International: West Conshohocken, PA, 702705 (2000)

  44. Torayca Carbon Fibres Technical Information. http://www.torayca.com/en/index.html

  45. Paiva, M.C., Bernardo, C.A., Edie, D.D.: A comparative analysis of alternative models to predict the tensile strength of untreated and surface oxidized carbon fibers. Carbon 39, 1091–1101 (2001)

    Article  Google Scholar 

  46. Lara-Curzio, E., Garcia, D.: Strength statistics of fiber bundles: the effect of fiber diameter variation along and among fibers, on the determination of the parameters of distribution of fiber strengths. Ceram. Eng. Sci. Proc. 22(3), 363–370 (2001)

    Article  Google Scholar 

  47. Prasanna Kumar, I., Mohite, P.M., Kamle, S.: Axial compressive strength testing of single carbon fibres. Arch. Mech. 65, 27–43 (2013)

    Google Scholar 

  48. Gupta, P.K.: Combined effect of flaw distribution and diameter variation on the statistics of glass fiber strength. J. Am. Ceram. Soc. 70(7), 486–492 (1987)

    Article  Google Scholar 

  49. Phoenix, S.L., Beyerlein, I.J.: Statistical strength theory for fibrous composite materials. Compr. Compos. Mater. 1, 559–639 (2000)

    Article  Google Scholar 

  50. Kaddour, A.S., Hinton, M.J.: Input data for test cases used in benchmarking triaxial failure theories of composites. J. Compos. Mater. 46(19–20), 2295–2312 (2012)

    Article  Google Scholar 

  51. Tagawa, T., Miyata, T.: Size effect on tensile strength of carbon fibers. Mater. Sci. Eng. A238, 336–342 (1997)

    Article  Google Scholar 

  52. Horst, R.: The Weibull distribution: A handbook, pp. 15–41. CRC Press, Fl, USA (2010)

    Google Scholar 

  53. Li, D.F., Wang, H.J., He, F., Wang, X.K.: Structure and properties of T300 and T700 carbon fibers. Carbon 45(6), 1379 (2007)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge the financial help provided by Aeronautical Research and Development Board (AR&DB), India, through the Grant under ACECOST, Phase III, to carry out this work.

Author information

Author notes

  1. R. Harikrishnan

    Present address: Department of Aerospace Engineering, Texas A&M University, College Station, 77840, USA

Authors and Affiliations

  1. Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, 208016, India

    R. Harikrishnan, P. M. Mohite & C. S. Upadhyay

Authors
  1. R. Harikrishnan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. M. Mohite
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. S. Upadhyay
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. M. Mohite.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Harikrishnan, R., Mohite, P.M. & Upadhyay, C.S. Generalized Weibull model-based statistical tensile strength of carbon fibres. Arch Appl Mech 88, 1617–1636 (2018). https://doi.org/10.1007/s00419-018-1391-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-018-1391-9

Keywords

Search

Navigation